 <p style="margin:0px;"></p><i><p style="margin:0px;">For courses in Transition to Advanced Mathematics or Introduction to Proof.</p></i><br><p style="margin:0px;"></p><b><p style="margin:0px;">Meticulously crafted, student-friendly text that helps build mathematical maturity </p><p style="margin:0px;"></p></b><p style="margin:0px;"><b><i>Mathematical Proofs: A Transition to Advanced Mathematics,</i> 4th Edition</b> introduces students to proof techniques, analyzing proofs, and writing proofs of their own that are not only mathematically correct but clearly written. Written in a student-friendly manner, it provides a solid introduction to such topics as relations, functions, and cardinalities of sets, as well as optional excursions into fields such as number theory, combinatorics, and calculus. The exercises receive consistent praise from users for their thoughtfulness and creativity. &nbsp;They help students progress from understanding and analyzing proofs and techniques to producing well-constructed proofs independently. This book is also an excellent reference for students to use in future courses when writing or reading proofs.</p><div><br></div>   

 <p style="margin: 0px;"></p><b><p style="margin: 0px;">0. Communicating Mathematics</p><p style="margin: 0px;"></p></b><p style="margin: 0px;"></p><blockquote><p style="margin: 0px;">0.1 Learning Mathematics</p><p style="margin: 0px;">0.2 What Others Have Said About Writing</p><p style="margin: 0px;">0.3 Mathematical Writing</p><p style="margin: 0px;">0.4 Using Symbols</p><p style="margin: 0px;">0.5 Writing Mathematical Expressions</p><p style="margin: 0px;">0.6 Common Words and Phrases in Mathematics</p><p style="margin: 0px;">0.7 Some Closing Comments About Writing</p></blockquote><p style="margin: 0px;"><b>1. Sets</b></p><p style="margin: 0px;"></p><blockquote><p style="margin: 0px;">1.1. Describing a Set</p><p style="margin: 0px;">1.2. Subsets</p><p style="margin: 0px;">1.3. Set Operations</p><p style="margin: 0px;">1.4. Indexed Collections of Sets</p><p style="margin: 0px;">1.5. Partitions of Sets</p><p style="margin: 0px;">1.6. Cartesian Products of Sets</p><p style="margin: 0px;">Chapter 1 Supplemental Exercises</p></blockquote><p style="margin: 0px;"><b>2. Logic</b></p><p style="margin: 0px;"></p><blockquote><p style="margin: 0px;">2.1. Statements</p><p style="margin: 0px;">2.2. The Negation of a Statement</p><p style="margin: 0px;">2.3. The Disjunction and Conjunction of Statements</p><p style="margin: 0px;">2.4. The Implication</p><p style="margin: 0px;">2.5. More On Implications</p><p style="margin: 0px;">2.6. The Biconditional</p><p style="margin: 0px;">2.7. Tautologies and Contradictions</p><p style="margin: 0px;">2.8. Logical Equivalence</p><p style="margin: 0px;">2.9. Some Fundamental Properties of Logical Equivalence</p><p style="margin: 0px;">2.10. Quantified Statements</p><p style="margin: 0px;">2.11. Characterizations of Statements</p><p style="margin: 0px;">Chapter&nbsp;2 Supplemental Exercises</p></blockquote><p style="margin: 0px;"></p><b><b><p style="margin: 0px;">3. Direct Proof and Proof by Contrapositive</p><p style="margin: 0px;"></p></b></b><p style="margin: 0px;"></p><blockquote><p style="margin: 0px;">3.1. Trivial and Vacuous Proofs</p><p style="margin: 0px;">3.2. Direct Proofs</p><p style="margin: 0px;">3.3. Proof by Contrapositive</p><p style="margin: 0px;">3.4. Proof by Cases</p><p style="margin: 0px;">3.5. Proof Evaluations</p><p style="margin: 0px;">Chapter&nbsp;3 Supplemental Exercises</p></blockquote><p style="margin: 0px;"></p><b><p style="margin: 0px;">4. More on Direct Proof and Proof by Contrapositive</p><p style="margin: 0px;"></p></b><p style="margin: 0px;"></p><blockquote><p style="margin: 0px;">4.1. Proofs Involving Divisibility of Integers</p><p style="margin: 0px;">4.2. Proofs Involving Congruence of Integers</p><p style="margin: 0px;">4.3. Proofs Involving Real Numbers</p><p style="margin: 0px;">4.4. Proofs Involving Sets</p><p style="margin: 0px;">4.5. Fundamental Properties of Set Operations</p><p style="margin: 0px;">4.6. Proofs Involving Cartesian Products of Sets</p><p style="margin: 0px;">Chapter&nbsp;4 Supplemental Exercises</p></blockquote><p style="margin: 0px;"></p><b><p style="margin: 0px;">5. Existence and Proof by Contradiction</p><p style="margin: 0px;"></p></b><p style="margin: 0px;"></p><blockquote><p style="margin: 0px;">5.1. Counterexamples</p><p style="margin: 0px;">5.2. Proof by Contradiction</p><p style="margin: 0px;">5.3. A Review of Three Proof Techniques</p><p style="margin: 0px;">5.4. Existence Proofs</p><p style="margin: 0px;">5.5. Disproving Existence Statements</p><p style="margin: 0px;">Chapter&nbsp;5 Supplemental Exercises</p></blockquote><p style="margin: 0px;"><b>6. Mathematical Induction</b></p><p style="margin: 0px;"></p><blockquote><p style="margin: 0px;">6.1 The Principle of Mathematical Induction</p><p style="margin: 0px;">6.2 A More General Principle of Mathematical Induction</p><p style="margin: 0px;">6.3 Proof By Minimum Counterexample</p><p style="margin: 0px;">6.4 The Strong Principle of Mathematical Induction</p><p style="margin: 0px;">Chapter&nbsp;6 Supplemental Exercises</p></blockquote><p style="margin: 0px;"><b>7. Reviewing Proof Techniques</b></p><p style="margin: 0px;"></p><blockquote><p style="margin: 0px;">7.1 Reviewing Direct Proof and Proof by Contrapositive</p><p style="margin: 0px;">7.2 Reviewing Proof by Contradiction and Existence Proofs</p><p style="margin: 0px;">7.3 Reviewing Induction Proofs</p><p style="margin: 0px;">7.4 Reviewing Evaluations of Proposed Proofs</p><p style="margin: 0px;">Chapter&nbsp;7 Supplemental Exercises</p></blockquote><p style="margin: 0px;"><b>8. Prove or Disprove</b></p><p style="margin: 0px;"></p><blockquote><p style="margin: 0px;">8.1 Conjectures in Mathematics</p><p style="margin: 0px;">8.2 Revisiting Quantified Statements</p><p style="margin: 0px;">8.3 Testing Statements</p><p style="margin: 0px;">Chapter&nbsp;8 Supplemental Exercises</p></blockquote><p style="margin: 0px;"><b>9. Equivalence Relations</b></p><p style="margin: 0px;"></p><blockquote><p style="margin: 0px;">9.1 Relations</p><p style="margin: 0px;">9.2 Properties of Relations</p><p style="margin: 0px;">9.3 Equivalence Relations</p><p style="margin: 0px;">9.4 Properties of Equivalence Classes</p><p style="margin: 0px;">9.5 Congruence Modulo n</p><p style="margin: 0px;">9.6 The Integers Modulo n</p><p style="margin: 0px;">Chapter&nbsp;9 Supplemental Exercises</p></blockquote><p style="margin: 0px;"><b>10. Functions</b></p><p style="margin: 0px;"></p><blockquote><p style="margin: 0px;">10.1 The Definition of Function</p><p style="margin: 0px;">10.2 The Set of All Functions From A to B</p><p style="margin: 0px;">10.3 One-to-one and Onto Functions</p><p style="margin: 0px;">10.4 Bijective Functions</p><p style="margin: 0px;">10.5 Composition of Functions</p><p style="margin: 0px;">10.6 Inverse Functions</p><p style="margin: 0px;">10.7 Permutations</p><p style="margin: 0px;">Chapter&nbsp;10 Supplemental Exercises</p></blockquote><p style="margin: 0px;"><b>11. Cardinalities of Sets</b></p><p style="margin: 0px;"></p><blockquote><p style="margin: 0px;">11.1 Numerically Equivalent Sets</p><p style="margin: 0px;">11.2 Denumerable Sets</p><p style="margin: 0px;">11.3 Uncountable Sets</p><p style="margin: 0px;">11.4 Comparing Cardinalities of Sets</p><p style="margin: 0px;">11.5 The Schr&ouml;der - Bernstein Theorem</p><p style="margin: 0px;">Chapter 11 Supplemental Exercises</p></blockquote><p style="margin: 0px;"><b>12. Proofs in Number Theory</b></p><p style="margin: 0px;"></p><blockquote><p style="margin: 0px;">12.1 Divisibility Properties of Integers</p><p style="margin: 0px;">12.2 The Division Algorithm</p><p style="margin: 0px;">12.3 Greatest Common Divisors</p><p style="margin: 0px;">12.4 The Euclidean Algorithm</p><p style="margin: 0px;">12.5 Relatively Prime Integers</p><p style="margin: 0px;">12.6 The Fundamental Theorem of Arithmetic</p><p style="margin: 0px;">12.7 Concepts Involving Sums of Divisors</p><p style="margin: 0px;">Chapter 12 Supplemental Exercises</p></blockquote><p style="margin: 0px;"><b>13. Proofs in Combinatorics</b></p><p style="margin: 0px;"></p><blockquote><p style="margin: 0px;">13.1 The Multiplication and Addition Principles</p><p style="margin: 0px;">13.2 The Principle of Inclusion-Exclusion</p><p style="margin: 0px;">13.3 The Pigeonhole Principle</p><p style="margin: 0px;">13.4 Permutations and Combinations</p><p style="margin: 0px;">13.5 The Pascal Triangle</p><p style="margin: 0px;">13.6 The Binomial Theorem</p><p style="margin: 0px;">13.7 Permutations and Combinations with Repetition</p><p style="margin: 0px;">Chapter 13 Supplemental Exercises</p></blockquote><p style="margin: 0px;"><b>14. Proofs in Calculus</b></p><p style="margin: 0px;"></p><blockquote><p style="margin: 0px;">14.1 Limits of Sequences</p><p style="margin: 0px;">14.2 Infinite Series</p><p style="margin: 0px;">14.3 Limits of Functions</p><p style="margin: 0px;">14.4 Fundamental Properties of Limits of Functions</p><p style="margin: 0px;">14.5 Continuity</p><p style="margin: 0px;">14.6 Differentiability</p><p style="margin: 0px;">Chapter 14 Supplemental Exercises</p></blockquote><p style="margin: 0px;"><b>15. Proofs in Group Theory</b></p><p style="margin: 0px;"></p><blockquote><p style="margin: 0px;">15.1 Binary Operations</p><p style="margin: 0px;">15.2 Groups</p><p style="margin: 0px;">15.3 Permutation Groups</p><p style="margin: 0px;">15.4 Fundamental Properties of Groups</p><p style="margin: 0px;">15.5 Subgroups</p><p style="margin: 0px;">15.6 Isomorphic Groups</p><p style="margin: 0px;">Chapter 15 Supplemental Exercises</p></blockquote><p style="margin: 0px;"><b>16. Proofs in Ring Theory (Online)</b></p><p style="margin: 0px;"></p><blockquote><p style="margin: 0px;">16.1 Rings</p><p style="margin: 0px;">16.2 Elementary Properties of Rings</p><p style="margin: 0px;">16.3 Subrings</p><p style="margin: 0px;">16.4 Integral Domains</p><p style="margin: 0px;">16.5 Fields</p><p style="margin: 0px;">Chapter 16 Supplemental Exercises</p></blockquote><p style="margin: 0px;"><b>17. Proofs in Linear Algebra (Online)</b></p><p style="margin: 0px;"></p><blockquote><p style="margin: 0px;">17.1 Properties of Vectors in 3-Space</p><p style="margin: 0px;">17.2 Vector Spaces</p><p style="margin: 0px;">17.3 Matrices</p><p style="margin: 0px;">17.4 Some Properties of Vector Spaces</p><p style="margin: 0px;">17.5 Subspaces</p><p style="margin: 0px;">17.6 Spans of Vectors</p><p style="margin: 0px;">17.7 Linear Dependence and Independence</p><p style="margin: 0px;">17.8 Linear Transformations</p><p style="margin: 0px;">17.9 Properties of Linear Transformations</p><p style="margin: 0px;">Chapter 17 Supplemental Exercises</p></blockquote><p style="margin: 0px;"><b>18. Proofs with Real and Complex Numbers (Online)</b></p><p style="margin: 0px;"></p><blockquote><p style="margin: 0px;">18.1 The Real Numbers as an Ordered Field</p><p style="margin: 0px;">18.2 The Real Numbers and the Completeness Axiom</p><p style="margin: 0px;">18.3 Open and Closed Sets of Real Numbers</p><p style="margin: 0px;">18.4 Compact Sets of Real Numbers</p><p style="margin: 0px;">18.5 Complex Numbers</p><p style="margin: 0px;">18.6 De Moivre&#39;s Theorem and Euler&#39;s Formula</p><p style="margin: 0px;">Chapter 18 Supplemental Exercises</p></blockquote><p style="margin: 0px;"><b>19. Proofs in Topology (Online)</b></p><p style="margin: 0px;"></p><blockquote><p style="margin: 0px;">19.1 Metric Spaces</p><p style="margin: 0px;">19.2 Open Sets in Metric Spaces</p><p style="margin: 0px;">19.3 Continuity in Metric Spaces</p><p style="margin: 0px;">19.4 Topological Spaces</p><p style="margin: 0px;">19.5 Continuity in Topological Spaces</p><p style="margin: 0px;">Chapter 19 Supplemental Exercises</p></blockquote><p style="margin: 0px;"></p><b><b><p style="margin: 0px;">Answers and Hints to Odd-Numbered Section Exercises</p><p style="margin: 0px;">References</p><p style="margin: 0px;">Index of Symbols</p><p style="margin: 0px;">Index</p></b></b><p style="margin: 0px;"></p><div><br></div>   

Gary Chartrand

Albert D. Polimeni

Ping Zhang

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